47 research outputs found
10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope
AbstractUsing oriented matroids, and with the help of a computer, we have found a set of 10 points inR4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4
Maximal admissible faces and asymptotic bounds for the normal surface solution space
The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint. The main results of this
paper are significant improvements upon the best known asymptotic bounds on the
number of admissible vertices, using polytopes in both the standard normal
surface coordinate system and the streamlined quadrilateral coordinate system.
To achieve these results we examine the layout of admissible points within
these polytopes. We show that these points correspond to well-behaved
substructures of the face lattice, and we study properties of the corresponding
"admissible faces". Key lemmata include upper bounds on the number of maximal
admissible faces of each dimension, and a bijection between the maximal
admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in
Journal of Combinatorial Theory A
Almost Quantum Correlations are Inconsistent with Specker's Principle
Ernst Specker considered a particular feature of quantum theory to be
especially fundamental, namely that pairwise joint measurability of sharp
measurements implies their global joint measurability
(https://vimeo.com/52923835). To date, Specker's principle seemed incapable of
singling out quantum theory from the space of all general probabilistic
theories. In particular, its well-known consequence for experimental
statistics, the principle of consistent exclusivity, does not rule out the set
of correlations known as almost quantum, which is strictly larger than the set
of quantum correlations. Here we show that, contrary to the popular belief,
Specker's principle cannot be satisfied in any theory that yields almost
quantum correlations.Comment: 17 pages + appendix. 5 colour figures. Comments welcom